31edo: Difference between revisions

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== Theory ==

31edo's ~~historical appeal is as the closed form of quarter-comma~~ [[~~meantone]], the regular tuning which splits a justly tuned 5/1 into four logarithmically equal parts representing~~ 3/2~~. This is because its~~ perfect fifth is flat of ~~the~~ just ~~interval [[3/2]]~~ by 5.2{{c}}, ~~which~~ is ~~approximately one-fourth~~ of ~~the syntonic comma~~ of ~~21.5 cents. Due to being close to~~ quarter-comma ~~meantone, 31edo also functions as [[septimal~~ meantone]]~~, as~~ [[7/4]] is just ~~a cent sharp of its augmented sixth~~. It is a very tone-efficient melodic approximation of the [[11-limit]] (and specifically the [[11-odd-limit]]), although it conflates [[9/7]] with [[14/11]] and [[11/8]] with [[15/11]]. Many [[7-limit]] JI scales are well-approximated in 31 (with tempering, of course). It also maps most [[15-odd-limit]] intervals [[consistent]]ly, the exceptions being [[13/9]], [[13/11]], and their [[octave complement]]s.

31edo's [[3/2|perfect fifth]] is flat of just by 5.2{{c}}, as befits a tuning of [[meantone]]. The major third is less than a cent sharp of just [[5/4]], making it slightly sharp of [[quarter-comma meantone]]. 31edo's approximation of [[7/4]], a cent flat, is also very close to just. It is a very tone-efficient melodic approximation of the [[11-limit]] (and specifically the [[11-odd-limit]]), although it conflates [[9/7]] with [[14/11]] and [[11/8]] with [[15/11]]. Many [[7-limit]] JI scales are well-approximated in 31 (with tempering, of course). It also maps most [[15-odd-limit]] intervals [[consistent]]ly, the exceptions being [[13/9]], [[13/11]], and their [[octave complement]]s.

Because of the near-just 5/4 and 7/4, 31edo is relatively quite accurate in the [[7-limit]]. Prime 11 is somewhat less accurate, making intervals like [[11/8]] off by about 9 cents, however intervals like [[11/9]] and [[11/6]] are approximated quite well because the errors cancel out. Other ways in which 31edo is especially accurate is that it represents a record in [[Pepper ambiguity]] in the 7-, 9-, and [[11-odd-limit]], which it is [[consistent]] through, and that it is the first [[trivial temperament|non-trivial]] edo to be consistent in the [[odd prime sum limit|11-odd-prime-sum-limit]]. It is also a prominent [[the Riemann zeta function and tuning #Zeta EDO lists|zeta peak edo]].

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 == Theory ==
-edo's historical appeal is as the closed form of quarter-comma [[meantone]], the regular tuning which splits a justly tuned 5/1 into four logarithmically equal parts representing 3/2. This is because its perfect fifth is flat of the just interval [[3/2]] by 5.2{{c}}, which is approximately one-fourth of the syntonic comma of 21.5 cents. Due to being close to quarter-comma meantone, 31edo also functions as [[septimal meantone]], as [[7/4]] is just a cent sharp of its augmented sixth. It is a very tone-efficient melodic approximation of the [[11-limit]] (and specifically the [[11-odd-limit]]), although it conflates [[9/7]] with [[14/11]] and [[11/8]] with [[15/11]]. Many [[7-limit]] JI scales are well-approximated in 31 (with tempering, of course). It also maps most [[15-odd-limit]] intervals [[consistent]]ly, the exceptions being [[13/9]], [[13/11]], and their [[octave complement]]s.
+edo's [[3/2|perfect fifth]] is flat of just by 5.2{{c}}, as befits a tuning of [[meantone]]. The major third is less than a cent sharp of just [[5/4]], making it slightly sharp of [[quarter-comma meantone]]. 31edo's approximation of [[7/4]], a cent flat, is also very close to just. It is a very tone-efficient melodic approximation of the [[11-limit]] (and specifically the [[11-odd-limit]]), although it conflates [[9/7]] with [[14/11]] and [[11/8]] with [[15/11]]. Many [[7-limit]] JI scales are well-approximated in 31 (with tempering, of course). It also maps most [[15-odd-limit]] intervals [[consistent]]ly, the exceptions being [[13/9]], [[13/11]], and their [[octave complement]]s.
 Because of the near-just 5/4 and 7/4, 31edo is relatively quite accurate in the [[7-limit]]. Prime 11 is somewhat less accurate, making intervals like [[11/8]] off by about 9 cents, however intervals like [[11/9]] and [[11/6]] are approximated quite well because the errors cancel out. Other ways in which 31edo is especially accurate is that it represents a record in [[Pepper ambiguity]] in the 7-, 9-, and [[11-odd-limit]], which it is [[consistent]] through, and that it is the first [[trivial temperament|non-trivial]] edo to be consistent in the [[odd prime sum limit|11-odd-prime-sum-limit]]. It is also a prominent [[the Riemann zeta function and tuning #Zeta EDO lists|zeta peak edo]].